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A358204 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)/(2*n)^n.

%I A358204 #16 Feb 16 2025 08:34:04
%S A358204 4,4,1,8,9,5,1,6,3,3,6,5,2,1,8,3,0,7,1,9,0,3,2,1,3,0,5,6,2,0,7,0,8,6,
%T A358204 3,7,8,7,4,7,9,9,2,8,4,7,4,3,6,9,4,8,0,4,7,7,8,3,7,8,7,0,3,9,0,7,0,7,
%U A358204 0,5,1,7,0,5,5,7,1,7,6,2,6,4,8,7,3,1,5,9,2,1,2,7,7,0,3,4,2,6,0,9
%N A358204 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)/(2*n)^n.
%H A358204 M. L. Glasser, <a href="https://doi.org/10.1080/00029890.2019.1565856">A note on Beukers's and related integrals</a>, Amer. Math. Monthly 126(4) (2019), 361-363.
%H A358204 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophomoresDream.html">Sophomore's Dream</a>.
%F A358204 Equals (1/2)*Integral_{x = 0..1} x^(x/2) dx.
%F A358204 Equals (-1/2)*Integral_{x = 0..1} log(x)*(x^(x/2)) dx.
%F A358204 Equals the double integral (1/2)*Integral_{x = 0..1, y = 0..1} (x*y)^(x*y/2) dx dy (apply Glasser, Theorem 1).
%e A358204 0.44189516336521830719032130562070863787479928...
%p A358204 evalf( add( (-1)^(n+1)/(2*n)^n, n = 1..50), 100);
%t A358204 RealDigits[N[Integrate[x^(x/2), {x, 0, 1}]/2, 120]][[1]] (* _Amiram Eldar_, Jun 21 2023 *)
%o A358204 (PARI) suminf(n=1, (-1)^(n+1)/(2*n)^n) \\ _Michel Marcus_, Nov 03 2022
%Y A358204 Cf. A073009, A098686, A358191, A358203.
%K A358204 cons,nonn,easy
%O A358204 0,1
%A A358204 _Peter Bala_, Nov 03 2022
%E A358204 a(98)-a(99) corrected by _Amiram Eldar_, Jun 21 2023